The value of 36(4^29^-1)(4^227^-1)(4^281^-1)(4^2243^-1) is:

Question

The value of $36(4\cos^29^\circ-1)(4\cos^227^\circ-1)(4\cos^281^\circ-1)(4\cos^2243^\circ-1)$ is:

Accepted Answer

Using $4\cos^2\theta-1 = \dfrac{\sin3\theta}{\sin\theta}$:

$$36 \cdot \frac{\sin27^\circ}{\sin9^\circ}\cdot\frac{\sin81^\circ}{\sin27^\circ}\cdot\frac{\sin243^\circ}{\sin81^\circ}\cdot\frac{\sin729^\circ}{\sin243^\circ} = 36\cdot\frac{\sin729^\circ}{\sin9^\circ}$$

Since $729^\circ = 720^\circ+9^\circ$, $\sin729^\circ = \sin9^\circ$, so the product telescopes to $36$.

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